Weakly Nonlinear Stability Analysis of Temperature/Gravity

Transp Porous Med (2012) 92:633–647DOI 10.1007/s11242-011-9925-4

Weakly Nonlinear Stability Analysisof Temperature/Gravity-Modulated StationaryRayleigh–Bénard Convection in a RotatingPorous Medium

B. S. Bhadauria · P. G. Siddheshwar ·Jogendra Kumar · Om P. Suthar

Received: 2 September 2011 / Accepted: 10 December 2011 / Published online: 28 December 2011© Springer Science+Business Media B.V. 2011

Abstract The effect of time-periodic temperature/gravity modulation on thermal instabil-ity in a fluid-saturated rotating porous layer has been investigated by performing a weaklynonlinear stability analysis. The disturbances are expanded in terms of power series of ampli-tude of convection. The Ginzburg–Landau equation for the stationary mode of convection isobtained and consequently the individual effect of temperature/gravity modulation on heattransport has been investigated. Further, the effect of various parameters on heat transporthas been analyzed and depicted graphically.

Keywords Ginzburg–Landau equation · Porous medium · Temperature modulation ·Gravity modulation · Rotation

List of Symbols

Latin SymbolsA Amplitude of streamline perturbationd Height of the fluid layerg Acceleration due to gravity

B. S. Bhadauria (B)Department of Applied Mathematics and Statistics, School for Physical Sciences,Babasaheb Bhimrao Ambedkar University, Lucknow 226025, UP, Indiae-mail: [email protected]

P. G. Siddheshwar · O. P. SutharDepartment of Mathematics, Bangalore University, Bangalore, Karnataka, Indiae-mail: [email protected]

O. P. Suthare-mail: [email protected]

B. S. Bhadauria · J. KumarDepartment of Mathematics, Faculty of Science, DST—Centre for InterdisciplinaryMathematical Sciences, Banaras Hindu University, Varanasi 221005, UP, Indiae-mail: [email protected]

123

634 B. S. Bhadauria et al.

kc Wavenumberη2 k2

c + π2

Nu Nusselt numberp Pressure

Pr Prandtl number,(

δνκT

)

Da Darcy number,(

Kd2

)

V a Vadasz number,( Pr

Da

)

Ra Rayleigh–Darcy number,(

βT g�T d KzδνκT

)

T a Taylor–Darcy number,(

2d2

δν

)

T Temperature�T Temperature difference across the fluid layert Time(x, y, z) Space co-ordinates

Greek Symbolsδ Porosityα Coefficient of thermal expansionκ Thermal diffusivityμ Dynamic viscosity

ν Kinematic viscosity,(

μρ0

)

ρ Fluid densityω Modulation frequencyδ1 Amplitude of temperature modulationδ2 Amplitude of gravity modulationφ Phase angleε Perturbation parameterτ τ = ε2t (small time scale)

Other Symbol

∇12 ∂2

∂x2 + ∂2

∂z2

Subscriptsb Basic statec Critical value0 Value of the un-modulated case

Superscripts′ Perturbed quantity∗ Dimensionless quantity

123

Weakly Nonlinear Stability Analysis 635

1 Introduction

The classical Rayleigh–Bénard convection due to bottom heating is widely known and is ahighly explored phenomenon. It has been extensively studied in a porous domain also (seeNield and Bejan 2006). It has numerous applications in many practical problems. A compre-hensive exposition on its applications in various fields is given in Nield and Bejan (2006),Ingham and Pop (2005) and Vafai (2005). The study of fluid convection in a rotating porousmedium is also of great practical importance in many branches of modern science, suchas centrifugal filtration processes, petroleum industry, food engineering, chemical engineer-ing, geophysics, and biomechanics. Several studies on the onset of convection in a rotatingporous medium have been reported (Friedrich 1983; Patil and Vaidyanathan 1983; Palm andTyvand 1984; Jou and Liaw 1987a,b; Qin and Kaloni 1995; Vadasz 1996a,b, 1998; Vadaszand Govender 2001; Straughan 2001; Desaive et al. 2002; Govender 2003).

In most studies related to thermal instability in a rotating porous medium saturated witha Newtonian fluid, steady temperature gradient is considered. However it is not so in manypractical problems. There are many interesting situations of practical importance in whichthe temperature gradient is a function of both space and time. This non-uniform temperaturegradient (temperature modulation) can be determined by solving the energy equation withsuitable time-dependent thermal boundary conditions and can be used as an effective mech-anism to control the convective flow. Innumerable studies are available which explain howa time-periodic boundary temperature affects the onset of Rayleigh-Bénard convection. Anexcellent review related to this problem is given by Davis (1976).

The study of Venezian (1969) or the Floquet theory has been extensively followed in thethermal convection problem in porous media when the boundary temperatures are time-periodic (Caltagirone 1976; Chhuon and Caltagirone 1979; Antohe and Lage 1996;Malashetty and Wadi 1999; Malashetty and Basavaraja 2002, 2003; Malashetty et al. 2006;Malashetty and Swamy 2007; Bhadauria 2007a,b; Bhadauria and Sherani 2008; Bhadauriaand Srivastava 2010). However, fewer studies are available on convection in a rotating porousmedia with temperature modulation of the boundaries [see Bhadauria 2007c,d; Bhadauriaand Suthar 2009 and references therein].

Another problem that leads to variable coefficients in the governing equations of ther-mal instability in porous media is the one involving vertical time-periodic vibration of thesystem. This leads to the appearance of a modified gravity, collinear with actual gravity,in the form of a time-periodic gravity field perturbation and is known as gravity modula-tion or g-jitter in the literature. Malashetty and Padmavathi (1997), Rees and Pop (2000,2001, 2003), Govender (2005a,b), Kuznetsov (2006a, b), Siddhavaram and Homsy (2006);Strong (2008a,b); Razi et al. (2009); Saravanan and Purusothaman (2009), Saravanan andArunkumar (2010), Malashetty and Swamy (2011), and Saravanan and Sivakumar (2010,2011) document some aspects of the problem.

The studies so far reviewed concern linear stability of the thermal system in a non-rotating/rotating porous media in the absence/presence of temperature/gravity modulation,and hence address only questions on onset of convection. If one were to consider heat andmass transports in porous media in the presence of temperature/gravity modulation, then thelinear stability analysis is inadequate, and the nonlinear stability analysis becomes inevita-ble. In the light of the above, we make a weakly nonlinear analysis of the problem using theGinzburg–Landau equation and in the process quantify the heat and mass transports in termsof the amplitude governed by the Ginzburg–Landau equation. The Ginzburg–Landau equa-tion has been solved numerically, and consequently the effect of modulation on the Nusseltnumber is studied.

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2 Mathematical Formulation

We consider a horizontal porous layer saturated with an incompressible viscous Boussinesqfluid confined between two parallel horizontal planes z = 0 and z = d . The porous layer isheated from below and cooled from above. A Cartesian frame of reference is chosen in sucha way that the origin lies on the lower plane and the z axis as vertically upward. The systemis rotating about z axis with a uniform angular velocity . The governing equations for thesystem are given by

∇ · �q = 0, (1)1

δ

∂ �q∂t

= − 1

ρ0∇ p + ρ

ρ0�g − 2

δ� × �q − ν

K�q, (2)

γ∂T

∂t+ 1

δ(�q · ∇)T = κT∇2T, (3)

ρ = ρ0 [1 − βT (T − T0)] , (4)

where �q is the velocity, ρ is the density of the fluid, p is hydrodynamic pressure, � =(0, 0, ) is the angular velocity, T is the temperature, and κT is the thermal conductivity ofthe fluid.

We assume the externally imposed boundary temperature to oscillate with time, accordingto the relations (Venezian 1969):

T = T0 + �T

2

[1 + ε2δ1 cos(ωt)

]at z = 0,

= T0 − �T

2

[1 − ε2δ1 cos(ωt + φ)

]at z = d,

(5)

where T0 is some reference temperature, and ω is the modulation frequency. The quantityε2δ1 is the amplitude of modulation. We consider small amplitude modulation such that bothε and δ1 are small.

We use the following scalings to non-dimensionalize the governing equations 1–3

(x, y, z)=d(x∗, y∗, z∗), (U, V, W )= δκT

d(U∗, V ∗, W ∗), t = d2

κTt∗, p= δμκT

Kp∗,

T = (�T ) · T ∗. (6)

The dimensionless form of the governing equations is given by

1

V a

∂q

∂t+ √

T a k̂ × q + q = −∇ p − RaDa

[1

βT �T− (T − T0)

]k̂, (7)

γ∂T

∂t+ (q · ∇) T = ∇2T . (8)

At the basic state, the fluid is assumed to be quiescent, and thus the basic state quantities aregiven by

ρ = ρb(z, t); p = pb(z, t); T = Tb(z, t); qb = (0, 0, 0). (9)

123


The basic state equations from (2)–(4) are

d pb

dz= −ρbg, (10)

γ∂Tb

∂t= ∂2Tb

∂2z. (11)

Solution of the above Eq. 11 subject to the non-dimensionalized thermal boundary conditions(5) is given by

Tb(z, t) = (1 − z) + ε2δ1 F(z, t), (12)

where

F(z, t) = Re[{

A(λ)eλz + A(−λ)e−λz} e−iωt],

A(λ) = 1

2

(e−iφ − e−λ

)(eλ − e−λ

) ; λ = (1 − i)

√ω

2. (13)

We now impose infinitesimal perturbations to the basic state in the form:

q = qb + Q′, T = Tb + T ′, ρ = ρb + ρ′, (14)

where primes denote the perturbation quantities. Now from Eq. 7, we have

1

V a

∂U

∂t− √

T aV + U = −∂p

∂x, (15)

1

V a

∂W

∂t+ W = −∂p

∂z+ RaT . (16)

From Eqs. 15 and 16, we eliminate pressure term to obtain an equation in the form:

1

V a

[∂

∂t

(∂U

∂z− ∂W

∂x

)]= −Ra

∂T

∂x, (17)

Also we have:1

V a

∂V

∂t+ √

T aU + V = 0, (18)

Introducing stream function � as U = ∂�/∂z and W = −∂�/∂x in the Eqs. 8, 17, and 18and using Tb in the reduced system, we get the equations as

(1

V a

∂∇2

∂t+ ∇2

)� − √

T a∂V

∂z+ Ra

∂T

∂x= 0, (19)

(1

V a

∂

∂t+ 1

)V + √

T a∂�

∂z= 0, (20)

(γ

∂

∂t− ∇1

2)

T = ∂ (�, T )

∂ (x, z)+ ∂�

∂x

[−1 + ε2δ1

∂ F

∂z

]. (21)

3 Stability Analysis

3.1 Linear Stability Analysis for Marginal Stationary Convection in Unmodulated System

To make this article self-contained, we give a brief account of important linear stability anal-ysis results. The linear stability results can be obtained by neglecting the nonlinear terms in

123


the above equations. The perturbed quantities are expanded by using normal mode techniqueas

�(x, z) = A sin(kcx) sin(π z)T (x, z) = B cos(kcx) sin(π z)V (x, z) = C sin(kcx) cos π z)

⎫⎬⎭ . (22)

In the above expressions, kc is the wave number. The Rayleigh number for stationary is givenby

Rast = η2(η2 + π2T a

)

kc2 , (23)

where η2 = π2 + k2c , and the solution of �, T and V is obtained as

�(x, z, t) = A sin(kcx) sin(π z)

T (x, z, t) = − kc

η2 A cos(kcx) sin(π z)

V (x, z, t) = −π√

T a

η2 A sin(kcx) cos(π z)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (24)

We note at this point that the oscillatory convection is possible (see Vadasz 1998), but wefocus our attention on only stationary convection in this article.

3.2 Nonlinear Stability Analysis

We use the time variations only at the slow time scale by considering τ = ε2t .⎡⎢⎢⎢⎢⎢⎣

ε2

V a

∂

∂τ∇1

2 − ∇12 Ra

∂

∂x−√

T a∂

∂z

−(

−1 + ε2δ1∂ F

∂z

)∂

∂xγ ε2 ∂

∂τ− ∇1

2 0

√T a

∂

∂z0

ε2

V a

∂

∂τ+ 1

⎤⎥⎥⎥⎥⎥⎦

×⎡⎣

�

TV

⎤⎦ =

⎡⎢⎢⎣

0∂(�, T )

∂(x, z)0

⎤⎥⎥⎦ . (25)

Now we use the following asymptotic expansion in Eqs. 19–21:

Ra = Ra0 + ε2 Ra2 + · · ·� = ε�1 + ε2�2 + · · ·T = εT1 + ε2T2 + · · ·V = εV1 + ε2V2 + · · ·

⎫⎪⎪⎬⎪⎪⎭

, (26)

where Ra0 is the critical Rayleigh number in unmodulated case. Substituting Eq. 26 in Eq. 25and comparing like powers of ε on both side, we get the solutions at different orders.

At first order:⎡⎢⎢⎢⎢⎣

−∇12 Ra0

∂

∂x−√

T a∂

∂z∂

∂x−∇1

2 0√

T a∂

∂z0 1

⎤⎥⎥⎥⎥⎦

⎡⎣

�1

T1

V1

⎤⎦ =

⎡⎣

000

⎤⎦ . (27)

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The above equations correspond to linear stability equations for stationary mode of convec-tion. The solutions of the above equations can be given by [using Eq. 24]

�1 = A(τ ) sin(kcx) sin(π z)

T1 = − kc

η2 A(τ ) cos(kcx) sin(π z)

V1 = −π√

T aA(τ ) sin(kcx) cos(π z)

⎫⎪⎪⎬⎪⎪⎭

, (28)

The system (27) gives the critical value of Rayleigh number and corresponding wave numberas obtained in the previous section.

At the second order, we have⎡⎢⎢⎢⎢⎣

−∇12 Ra0

∂

∂x−√

T a∂

∂z∂

∂x−∇1

2 0√

T a∂

∂z0 1

⎤⎥⎥⎥⎥⎦

⎡⎣

�2

T2

V2

⎤⎦ =

⎡⎣

R21

R22

R23

⎤⎦ , (29)

where

R21 = 0, (30)

R22 = ∂(�1, T1)

∂(x, z)= −πk2

c

2η2 [A(τ )]2 sin(2π z), (31)

and

R23 = 0. (32)

One can obtain second-order solutions as

�2 = 0

T2 = − k2c

8πη2 [A(τ )]2 sin(2π z)

V2 = 0

⎫⎪⎪⎬⎪⎪⎭

. (33)

The horizontally averaged Nusselt number Nu(τ ) for the stationary mode of convection (thepreferred mode in this problem) is given by

Nu(τ ) =[

kc2π

∫ 2π/kcx=0 (1 − z + T2)zdx

]z=0[

kc2π

∫ 2π/kcx=0 (1 − z)zdx

]z=0

. (34)

Substituting Eq. 33 in Eq. 34 and simplifying, we get

Nu(τ ) = 1 + k2c [A(τ )]2

4η2 . (35)

At the third order, we have⎡⎢⎢⎢⎢⎣

−∇12 Ra0

∂

∂x−√

T a∂

∂z∂

∂x−∇1

2 0√

T a∂

∂z0 1

⎤⎥⎥⎥⎥⎦

⎡⎣

�3

T3

V3

⎤⎦ =

⎡⎣

R31

R32

R33

⎤⎦ , (36)

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where

R31 = −Ra2∂T1

∂x− 1

V a

∂(∇2�1)

∂τ, (37)

R32 = ∂�1

∂x

∂T2

∂z+ δ1

∂�1

∂x

∂ F

∂z− γ

∂T1

∂τ, (38)

R33 = − 1

V a

∂V1

∂τ. (39)

Substituting �1, T1, T2, V1, and V2 from Eqs. 28 and 33 in Eqs. 37–39, we get

R31 =[

η2

V a

dAdτ

− R2k2

c

η2 A]

sin(kcx) sin(π z), (40)

R32 =[γ

kc

η2

dAdτ

+ δ1∂ F

∂zkcA − k3

c A3

4η2 cos(2π z)

]cos(kcx) sin(π z), (41)

R33 = π√

T a

V aη2

dAdτ

sin(kcx) cos(π z). (42)

The solvability condition for the existence of the third-order solution is given by

1∫

z=0

2π/kc∫

x=0

[�̂1 R31 + Ra0T̂1 R32 − V̂1 R33]dxdz = 0, (43)

where �̂1, T̂1, and V̂1 are the solutions of the adjoint system of the first order, given by

�̂1 = −A(τ ) sin(kcx) sin(π z)

T̂1 = − kc

η2 A(τ ) cos(kcx) sin(π z)

V̂1 = π√

T a

η2 A(τ ) sin(kcx) cos(π z)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (44)

Using Eqs. 40–42 and 44 in the Eq. 43 and simplifying, we get the Ginzburg–Landau equationfor stationary instability with a time-periodic coefficient in the form:

(1

V a+ γ

η2

)dA(τ )

dτ− F(τ )A(τ ) +

[k2

c

8η2 + π4 k2c T a

2V a2(η2 + π2 T a

)]

A3(τ ) = 0, (45)

where

F(τ ) = Ra2

Ra0− δ2I(τ ), (46)

and

I(τ ) =1∫

0

[∂ F(z, τ )

∂zsin2(π z)

]dz. (47)

The solution of Eq. 45, subject to the initial condition A(0) = a0 where a0 is a chosen initialamplitude of convection, can be obtained by means of a numerical method. In calculations,we may assume Ra2 = Ra0 to keep the parameters to the minimum.

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4 Time-Periodic Gravity field

Now we study the effect of a periodically varying gravitational field. For this, we assume

g = g0(1 + ε2δ2 cos(ωt)), (48)

where g0 is some reference value of the gravitational force, ω is the frequency of modulation,τ is the slow time scale defined in the previous section, and the gravitational force is givenby �g = (0, 0, −g). It should be noted here that the basic temperature in the present case isconsidered to be steady, and is given by

Tb = 1 − z. (49)

Using above expression in the Eqs. 19–21, we get the equation corresponding to gravity mod-ulation. The linear stability analysis can be performed exactly in the same way as describedin the above section, and the same results (Eqs. 23 and 35) can be readily drawn.

Further the system (25) reduces to the form:⎡⎢⎢⎢⎢⎢⎣

ε2

V a

∂

∂τ∇1

2 − ∇12 Ra[1 + ε2δ2 cos(ωt)] ∂

∂x−√

T a∂

∂z∂

∂xγ ε2 ∂

∂τ− ∇1

2 0

√T a

∂

∂z0

ε2

V a

∂

∂τ+ 1

⎤⎥⎥⎥⎥⎥⎦

×⎡⎣

�

TV

⎤⎦ =

⎡⎢⎢⎣

0∂(�, �)

∂(x, z)0

⎤⎥⎥⎦ . (50)

Also, we obtained here the same result at the first and second orders as reported in Eqs. 29and 34. In particular, the Nusselt number for stationary mode of convection can be obtainedas

Nu(τ ) = 1 + k2c [A(τ )]2

4η2 . (51)

At the third order, the expression in Eq. 30, in this case, are obtained as

R31 = −[Ra2 + Ra0δ2 cos(ωτ)]∂T1

∂x− 1

V a

∂(∇12�1)

∂τ, (52)

R32 = ∂�1

∂x

∂T2

∂z− γ

∂T1

∂τ, (53)

R33 = − 1

V a

∂V1

∂τ. (54)

For the above given values of R31, R32, and R33, the solvability condition (Eq. 44) producesthe Ginzburg–Landau amplitude equation given by

(1

V a+ γ

η2

)dA(τ )

dτ− F(τ )A(τ ) +

[k2

c

8η2 + π4 k2c T a

2V a2(η2 + π2 T a

)]

A3(τ ) = 0, (55)

where

F(τ ) = Ra2

Ra0+ δ2 cos(ωτ). (56)

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5 Results and Discussions

In this article, we have studied the effect of temperature and gravity modulations on thermalinstability in a rotating fluid saturated porous layer. A weakly nonlinear stability analysis hasbeen performed to investigate the effect of temperature/gravity modulation on heat transport.The temperature modulation has been considered in the following three cases:

1. In-phase modulation (IPM) (φ = 0),2. Out-phase modulation (OPM) (φ = π ) and3. Lower-boundary modulated only (LBMO) (φ = −i∞),

The effect of temperature modulation has been depicted in Figs. 1, 2, 3 while that of gravitymodulation in Fig. 4. From the Figs. 1, 2, 3, we observe the following general result forNusselt number Nu:

NuIPM < NuLBMO < NuOPM.

The parameters that arise in the problem are T a, V a, φ, δ1, and δ2 and these parametersinfluence the convective heat transports. The first two parameters relate to the fluid and thestructure of the porous medium, and the last three concern the two external mechanisms ofcontrolling convection.

Due to the assumption of low porosity medium, the values considered for V a are 100, 150and 200. Because small amplitude modulations are considered, the values of δ1 and δ2 liebetween 0 and 0.5. Further, the modulation of the boundary temperature and the verticaltime-periodic fluctuation of gravity are assumed to be of low frequency. At low range offrequencies, the effect of frequencies on onset of convection as well as on heat transport isminimal. This assumption is required to ensure that the system does not pick up oscillatory

0.02, 0.05, 0.08

Va 100, Ta 60, 2

0.00 0.05 0.10 0.15 0.20

1.0

1.5

2.0

2.5

Nu 1, 2, 5

0.05, Ta 60, Va 100

0.00 0.05 0.10 0.15 0.20

1.0

1.5

2.0

2.5

Nu

Ta 3060

100

0.05, Va 100, 2

0.00 0.05 0.10 0.15 0.20

1.0

1.5

2.0

2.5

Nu Va 100

150200

0.05, Ta 60, 2

0.00 0.05 0.10 0.15 0.20

1.0

1.5

2.0

2.5

3.0

Nu

(a)

(c) (d)

(b)

Fig. 1 Variation of Nusselt number Nu with time τ for different values of (a) δ1, (b) ω, (c) T a, (d) V a

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Va 100, Ta 60, 2

0.02 0.050.08

0 5 10 15 202.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Nu

0.05, Ta 60, Va 100

1 2 5

0 5 10 15 202.0

2.1

2.2

2.3

2.4

2.5

2.6

Nu

0.05, Va 100, 2

Ta 30 60 100

0 5 10 15 20

1.0

1.5

2.0

2.5

Nu

0.05, Ta 60, 2

Va 100150 200

0 5 10 15 20

1.0

1.5

2.0

2.5

3.0

Nu

(a) (b)

(c) (d)


0.02 0.05 0.08

Va 100, Ta 60, 2

0 5 10 15 202.2

2.3

2.4

2.5

2.6

Nu

1 2 5

0.05, Ta 60, Va 100

0 5 10 15 202.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

Nu

Ta 30 60 100

0.05, Va 100, 2

0 5 10 15 20

1.0

1.5

2.0

2.5

Nu Va 100

150 200

0.05, Ta 60, 2

0 5 10 15 20

1.0

1.5

2.0

2.5

3.0

Nu

(a)

(c) (d)

(b)


convective mode at the onset due to modulation in a situation that is conducive otherwise tostationary mode.

In Fig. 1a–d, we have plotted the Nusselt number Nu with respect to time τ for the caseof IPM. From the figures, we find that for small time τ , the value of Nu does not alter and

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0.02 0.050.08

Va 100, Ta 25, 2

0 5 10 15 202.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Nu

1 2 5

0.05, Ta 25, Va 100

0 5 10 15 202.0

2.1

2.2

2.3

2.4

2.5

2.6

Nu

Ta 30 60 100

0.05, Va 100, 2

0 5 10 15 20

1.0

1.5

2.0

2.5

3.0

Nu Va 100 150

200

0.05, Ta 60, 2

0 5 10 15 20

1.0

1.5

2.0

2.5

3.0

Nu

(a) (b)

(d)(c)


remains almost constant, then it increases on increasing τ , and finally becomes steady onfurther increasing the time τ .

From the Fig. 1a, b, we observed that on increasing amplitude of modulation δ1 and fre-quency of modulation ω, the value of Nusselt number Nu does not alter. Thus, the effects ofincreasing δ1 and ω have negligible effect on rate of heat transfer in case of IPM. From Fig. 1c,it has been examined that on increasing Darcy–Taylor number T a, the value of Nusselt num-ber decreases, thus decreasing the rate of heat transport and hence stabilizing the system.The effect of Vadasz number V a on Nu is observed in Fig. 1d. From the figure, we find thatNu increases with increasing V a, thus advancing the convection. Further, we observed thatthe results obtained in this case are qualitatively similar to those of unmodulated case.

Figure 2a–d shows the plots of Nu with time τ for the case of OPM. From Fig. 2a, we findthat the effect of increasing modulation amplitude δ1 on Nu is to increase the magnitude ofNu, i.e., rate of heat transport increases. Figure 2b shows that, on increasing the frequency ofmodulation ω, the magnitude of Nu does not change, but wavelength of oscillations becomeshorter with increasing ω. From the Fig. 2c, we observe that on increasing T a, the value ofNu decreases, thus the effect of increasing T a is to decrease the rate of heat transfer, andhence to suppress the convection. Figure 2d displays the effect of V a on Nu. From the figure,we examine that on increasing V a, Nu increases, therefore, the effect of increasing V a isto increase the rate of heat transfer across the porous layer.

It is obvious from the figure that

Nu/δ1=0.02 < Nu/δ1=0.05 < Nu/δ1=0.08.

In Fig. 3a–d, we depict the variation of Nusselt number Nu with time τ for lower-boundary temperature modulation only. From the figure, we find qualitatively similar resultsto that in Fig. 2a–d.

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In Fig. 4a–d, we depict the variation of Nu with respect to slow time τ under gravity mod-ulation. From Fig. 4a, we find that an increment in amplitude of modulation δ2, increasesNu; however, the wavelength of oscillations remains unaltered. From Fig. 4b, which showsthe effect of frequency of modulation ω on Nu, we observe that on increasing ω, the valueof Nu remains unchanged wherein the wavelength of oscillations decreases. The effect ofrotation has been shown in Fig. 4c, and is found to have stabilizing effect. However, fromFig. 4d, we observe that on increasing Vadasz number V a, the value of Nu increases, there-fore advancing the convection. Further, the results in Fig. 4a–d are found qualitatively similarto that of Figs. 2a–d and 3a–d.

6 Conclusion

In the present study, the effects of temperature/gravity modulation in a rotating fluid sat-urated porous layer have been analyzed by performing nonlinear stability analysis usingGinzburg–Landau equation. On the basis of the above discussion following conclusions aredrawn:

(1) NuIPM < NuLBMO < NuOPM for temperature modulation.(2) The effect of increasing Taylor number T a on Nu is to decrease the heat transport.(3) The effect of increasing Vadasz number V a is to advance the convection for in-phase

and out-phase as well as for the case when only lower plate is modulated.(4) On increasing the magnitude of modulation and frequency of modulation for the case

of in-phase temperature modulation, we find negligible effect on Nu, while there issignificant effect in the cases of OPM and LBMO.

(5) The results of gravity modulation are found qualitatively similar to the cases of OPMand LBMO.

Acknowledgements Part of this study was done during the lien period sanctioned by Banaras Hindu Univer-sity, Varanasi, India to the author BSB to work as Professor of Mathematics at Babasaheb Bhimrao AmbedkarUniversity, Lucknow. The authors are grateful to the referees for their most useful comments.

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Weakly Nonlinear Stability Analysis of Temperature/Gravity